STOCHASTIC TECHNIQUE FOR SOLUTIONS OF NONLINEAR FIN EQUATION ARISING IN THERMAL EQUILIBRIUM MODEL

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1 STOCHASTIC TECHNIQUE FOR SOLUTIONS OF NONLINEAR FIN EQUATION ARISING IN THERMAL EQUILIBRIUM MODEL Iftikhar AHMAD, Hina QURESHI, Muhammad BILAL 3, and Muhammad USMAN 4* Dpartmnt of Mathmatics, Univrsity of Gujrat, Pakistan Dpartmnt of Mathmatics, Univrsity of Gujrat, Pakistan 3 Dpartmnt of Mathmatics, Univrsity of Pahang, Malaysia 4* Dpartmnt of Mathmatics, Univrsity of Dayton, USA * Corrsponding author; musman@udayton.du In this study, a stochastic numrical tchniqu is usd to invstigat th numrical solution of th hat transfr tmpratur dpndnt systm using Fd-forward artificial nural ntworks (ANN). Th Mathmatical modl of Fin quation is formulatd with th hlp of ANN. Th ffct of th hat on a rctangular fin with thrmal conductivity and tmpratur dpndnt intrnal hat gnration is calculatd through nural ntworks optimization with optimizrs lik activ st tchniqu (AST), intrior point tchniqu (IPT), pattrn sarch (PS), gntic algorithm (GA) and a hybrid approach of PS-IPT, GA-AST, GA-IPT and GA-SQP with diffrnt slctions of wights. Th govrning fin quation is transformd into an quivalnt nonlinar scond ordr ordinary diffrntial quation. For this transformd ordinary diffrntial quation modl w hav prformd svral simulations to provid th justification of bttr convrgnc of rsults. Morovr, th ffctivnss of th dsignd modl is validatd through a complt statistical analysis. This study rvals th importanc of rctangular fins during th hat transformation through th systm. Kywords: Hat distribution, thrmal conductivity, gntic algorithm (GA), Intrior point algorithm (IPT), Activ st tchniqu (AST). Introduction Hat transfr is th convrsion of nrgy bcaus of tmpratur diffrncs. Whn two bodis having diffrnt tmpraturs com in contact, transfr of hat taks plac btwn thm until thrmal quilibrium is attaind []. For th consistnt working of lctronics with high nrgy dnsity, it is ncssary to formulat ffctiv cooling schms []. Fr lctrons motion within smiconductors causs not only xcssiv hat loss but also gnrats th signal nois [3]. Almost vry industrial structur is dsignd to work within dfinit tmpratur limits, going byond ths limits by ovrhating causs systm failur. To avoid this problm of xcssiv hat gnration, lctronic componnts hav to pass through a complx ntwork of hat rsistanc [4]. Th following diffrnt augmntation mthods can b usd to incras hat transfr; Compound mthods, activ mthods and passiv mthods [3]. Passiv cooling mthods ar widly prfrrd for lctronic and powr dvics, as thy ar low cost, noislss and asy to us. Incras in hat transfr by passiv tchniqus can b attaind by using; Tratd surfacs, rough surfacs and xtndd surfacs (fins). Extndd surfacs dlivr fficint hat transfr xpansion

2 through incrasing th surfac ara. In th xtndd surfac (fins) modrn improvmnts hav bn mad to amnd finnd surfacs that also tnd to improv th hat transfr cofficints [3]. Fins ar xtndd surfacs usd in an attmpt to incras th hat transfr rat [5]. Both hat conduction and convction tak plac simultanously in this cas. Convction occurs outsid th surfac with insid conduction [6]. Rcntly, many rsarchrs hav bn contributd to hat transfr, prssur loss and thrmal prformancs of diffrnt fins [7]. Fins ar commonly usd in many manufacturing and nginring systms such as in air conditioning systms, turbins, lctronic apparatuss and cooling of lctronic componnts [8]. Fins ar in diffrnt shaps such as rctangular, circular, pin fin rctangular and pin fin triangular, but bcaus of its simpl dsign, low production costs and its convnint constructing procss th rctangular fin is broadly usd as compar to othrs [9]. Fins utilization is on of an conomical way out which can dissipat undsird hat and it has bn ffctivly usd for many manufacturing uss. Bcaus of th scintific significanc and wid usag of fins, a nonlinar diffrntial quation of hat conduction is countd. Aziz and Na considrd a nonlinar fin quation subjct to variabl thrmal conductivity [0]. Chung and Iyr xamind th optimization difficultis for longitudinal fins []. Kraus, Aziz and Wlty studid many wll-known mathmatical xampls which dscrib th hat transfr in diffrnt typs of fins with variabl boundary conditions [5]. Khani and Aziz usd th homotopy mthod to stablish an analytical solution for th thrmal prformanc of a straight fin of a trapzoidal profil whn both th thrmal conductivity and th hat transfr cofficint ar tmpratur dpndnt []. A dcomposition tchniqu for solving longitudinal fins was xtnsivly rviwd by Ching and Cha'o [3]. Min rportd a dcomposition mthod in ordr to xamin th thrmal ffcts rctangular fins [6]. Sin t al. summarizd an stimatd solution for fins subjct to thrmal conductivity [4]. Yu t al. usd a doubl dcomposition schm for solving th oscillating bas tmpratur in longitudinal fins [5]. B. Kundu and D. Bhanja rviwd th optimization study of a constructal fin with thrmal conductivity [6]. Abdul and Tigang calculatd substitut solutions for rctangular fins [7]. K. Hossini t al. applid th homotopy mthod for th solution of fin subjct to intrnal hat that is tmpratur dpndnt [8]. Rohit and Rajin usd Adomian dcomposition mthodology to solv nonlinar rctangular fin problm [9]. Jun t al., utilizd an application of th modifid dcomposition mthod for nonlinar straight fins [0]. Th proposd stochastic numrical tchniqus hav th som advantags on othrs rportd tchniqus: firstly, this tchniqu has global natur for finding th proposd approximatd solutions bsid it has ability to gt th solution of linar and nonlinar problms of physical and mathmatical scincs ovrall; scondly, it provids good accurat rsults and which ar continuous ovr whol domain of nonlinar systm; thirdly, in th application sns it has global approximation and can find th rquird drivativs of Fin-quation in th whol domain for solution; fourthly, this tchniqu did not affct by th numrical computd rsults round off rrors. In th prsnt study, nw computational intllignc tchniqus ar prsntd for solving fins. Artificial intllignc tchniqus basd on nural ntworks optimizd with fficint global and local sarch mthodologis hav bn xtnsivly usd to solv a varity of th linar and nonlinar systms basd on ordinary and partial diffrntial quations [-]. Morovr, on-dimnsional fin problm solution and triangular fins ffcts hav bn studid [3], th thrmal dsign of a countr

3 flow hat xchangr using air as th working fluid is invstigatd [4]. Furthr, som latst work has bn don rcntly for intrst s th rfrncs [5-8]. Transformation Function Considr a fin of rctangular profil, having hat gnration ( q ), thrmal conductivity ( k ), primtr ( P ), lngth ( L ) and sction ara ( A ), which is attachd to a surfac of constant T b tmpratur ( ). Th fin loss hat to th surrounding nvironmnt, ( ). This loss of hat is through a constant cofficint ( h ) of convction hat transfr Hr w assumd hat transfr occurrd only in th longitudinal x-dirction. 0, d dt hp k T T 0 q dx dx A dt X 0, 0, X L, T T, () dx b also k k T T q q T T [ ] & [ ], whr is th thrmal conductivity and rprsnts intrnal hat gnration whn th surrounding tmpratur is whil paramtrs signifying hat gnration and thrmic conductivity variation [8,9]. Hr through variabl transformation, w will chang th govrning quation of th problm. Considring a nw rlation X T T t, 0 L T T b 0 Substituting ths valus of Eq. (3) into Eq. () w can obtain d N N G d 0, dt C dt G d t 0, 0, t,, dt k 0 T 0 T 0 () q 0 and ar th (3) (4) (5) whr, various dimnsionlss trms ar mntiond blow, X T T t, T T, L C 0 b 0, T T b 0 N L hp, Ak 0 q A G 0, hpt T b 0 G T T b 0. Mathmatical Modlling In this sction, w prsntd mathmatical modling of rctangular fin quation, through artificial nural ntworks (ANN). A mathmatical modl of th fin quation is formulatd with th hlp of a fd-forward ANN. In th nural ntwork modl, th nd drivativ can b approximatd by applying an activation function (log sigmoid function) [9-30]. 3

4 N () t i i ( ) i t i ( t ) N i i d dt i i i ( t ) ( i i ) ( t ) ( t ) d N i i i i, i i i dt ( t ) 3 ( t ) ( i i ) ( i i ) whr hat symbol rprsnts stimatd valus, m dnots th numbr of nurons and i, ar optimization wights or adjustabl paramtrs. (6) i and Fitnss Function Th fitnss function, E, for th modl has bn constructd in an unsuprvisd mannr as th sum of th rror du to trms prsnt in th quation and th rror in boundary conditions is dfind as E E E. Th rror function rlatd to transformd fin Eq. (4) is writtn as K [( )( ) ( ) E ( )] c c N N G c, K j 0 j j j j (7) j whr t ( t0 0, t, t,..., t j K) with stp siz, whr rror trm rlatd to th boundary conditions Eq. (5) is dfind as h E i E ( ) ( ) 0. (8) Larning Tchniqus Th Pattrn Sarch (PS) tchniqu blongs to th class of optimization algorithms that do not rquir th gradint of th function undr considration. Hook and Jvs wr th first to introduc th PS mthod [3], howvr, th convrgnc of th PS tchniqu was stablishd by Yu [3]. In th PS tchniqu, a squnc of points that approach an optimal point is computd. In ach stp, th schm sarchs a st of points calld a msh around th optimal point of th prvious stp. Th msh is cratd by adding th currnt point to a scalar multipl of a st of vctors calld a pattrn [33]. If th pattrn sarch algorithm finds a point in th msh that improvs th objctiv function at th currnt point, th nw point bcoms th currnt point at th nxt stp of th algorithm. Optimization Stps : Initialization: Adjust th solvr according to th ncssity of th problm. Initializ th paramtr s valus in accordanc with an optimal tchniqu from th tabl of paramtr stting as shown in Tabl.. Vctor W (,,..., m,,,..., m,,,..., m) act as a straight point for solvr hr m signifis th numbr of nurons. By continuous rlations, w can stimat scond and third ordr drivativ of th solution t. : Fitnss Evaluation: For optimization built-in function in MATLAB is invokd. In optimization tool prss start button in ordr to start th optimization procdur of problm. Now for fitnss valuation of th problm, th procss starts until th 4

5 trmination critria achiv. 3: Trmination Critria: If any on of th following critria achivd, trminat th implmntation of th solvr. a) Rquird lvl of prsnt fitnss achivd as shown in Tabl. b) Th total numbr of itration accomplishd. c) Th prdfind fitnss valu is obtaind i.., E : Storag: Sav th optimal wights (variabls) and fitnss valus and computational tim takn by th algorithm. 5: Statistical Analysis: Rpat th procss from to 4 for satisfactorily larg numbr of tims. A schmatic structur of th fin systm with diffrnt layrs is shown in Fig. Fig.: Schmatic Structur of th Fin Systm Numrical Rsults In this sction hat transfr analysis of solid fin with a rctangular profil having hat gnration and air inlt flow is prsntd using diffrnt proposd mthodologis. Th optimizrs succssfully run with MATLAB R03(b) softwar packag to find th rquird numrical rsults, using Windows XP, with a prsonal computr having an Intl (R) Cor (TM) Duo CPU (3.33 GHz) and.00 GB RAM (3.9 GHz). Th rsults ar formulatd from Eq. (7) and Eq. (8) through ANNs with th stting of 30 wights in which training st is dividd into 0 qual spac stps btwn t [0,] with stp siz h=0.. Th paramtrs stting ar availabl in Tabl which usd in optimtool. Proposd rsults for diffrnt solvrs as listd in Tabl as θ AST, θ IPT, θ PS, θ GA, θ PS IPT, yilding th fitnss valu,.60e 0,.4E-0, 8.9E-07, 8.74E-06,.9E-09 rspctivly. In ordr to prov th applicability and ffctivnss of th proposd schm a non-linar homognous fin problm has bn considrd. Tab. : Paramtr valus for Activ St tchniqu, Pattrn Sarch and Gntic Algorithms. AST PS GA Paramtr Valus Paramtr Valus Paramtr Valus Start Point Cration Randn(,30) Max Itration Pop Init Rang (-,) Max. Prturbation 0. Max Fun Evals Population Siz Min. Prturbation E-0 Tol Fun E-35 Bounds (-30,30) Max Itration 0000 Tol Con E-3 Slction Fun Stoc Uni Max Fun Evals 5000 Tol Msh E-33 Gnration 000 Function Tolranc E-3 X Tolranc E-37 Tol Fun E-8 Tol Con E-3 Initial Siz.0 StallGn Limit 000 Fitnss Limit E-37 Pnalty Fact 00 Tol Con E-7 X Tolranc E-35 Bind Tolr E-3 Fitnss Limit E-5 5

6 Tab. : Rportd rsult with rfrnc (xact) rsult t () t AST IPT PS GA PSIPT AST () t (.0948t.9787 ) ( t ) ( t 0.086) (.0375t.667) ( t ) (.39539t.498) (.4359t.4468) ( t ) ( 0.875t 0.30) (.506t.3949) (9) IPT () t (.36043t.9675) ( 0.839t 0.70) ( t 0.583) ( t.58036) ( t 0.73) (.4479t.77447) (.377t.48089) ( t 0.474) ( 0.093t 0.488) ( 0.907t.35697) (0) 6

7 GA() t ( 0.573t ) (.8903t.69836) (.797t.5066) (.38938t ) ( t ) ( ) t ( t.9373) ( t.48799) (.8005t.5038) (.47336t.35880) () Th solutions ar givn in th abov quation rmaind valid for th ntir domain [0,]. Th valus of approximat rror AE for th solution ar dtrmind for proposd algorithms and rsults ar providd in Tab. 3 for AST, IPT, PS, GA, PS-ITP lis in th rang 0 0, 0 0, , 0 0 and 0 0, rspctivly. Tab. 4 shows th comparison of diffrnt proposd rsults of absolut rror (AE). Th man valu of AE (MAE) is calculatd in th intrval [0,]. Rsults of MAE for AST, IPT, PS, GA, PS-IPT ar dtrmind rspctivly as 4.0E 06, 4.04E 07,.65E 04, 4.67E 07 and 6.93E 03. Rsults of MSE for AST, IPT, PS, GA, PS-IPT ar dtrmind rspctivly as.5e 4,.09E, 9.99E 08, 5.8E 05, and.9e. Th rliability and slf-ffctivnss of th solutions providd by stochastic schms can only b validatd through Mont Carlo simulations and its comprhnsiv statistical analysis. In this rgard, 00 indpndnt runs ar carrid out for ach solvr as shown in Fig.. Tab. 3: Approximatd Errors rportd by th xact rsult takn from diffrnt algorithms. t θ rf θ AST θ rf θ IPT θ rf θ PS θ rf θ GA θ rf θ Ps IPT E E E-05.93E-05.0E E-07.3E E E-06.96E E-07.86E-07.0E E E E-07.73E E E E E E-08.64E E E E E-07.93E E E E-07.93E-07 4.E E E E-07.33E E E-08.58E-07.68E E E-07.60E-07.05E E E-0.4E-0 8.9E E-09 7

8 Tab. 4: Comparison of Absolut rrors of diffrnt algorithms T AE (GA-30) AE (GA-AST) AE (GA-IPT) AE (GA-SQP) AE (GA-45) AE (GA-60) AE (PS-IPT) E-03.07E-05.9E E E E E-03.07E-05.E E E E E E-06.75E E E E E E-06.4E E E E E E-06.9E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-03.58E E-07.78E E E E-03.33E-06.E-07.95E E E-09 Fig. : Shows th MAE, MSE and RMSE takn by 00 indpndnt runs. Statistical analysis In ordr to find small diffrncs among rsults of AE, w prsntd statistical analysis particularly fitting of normal distribution basd on absolut rrors (AEs) of AST, IPT, PS, GA, hybrid PS-IPT, GA-AST, GA-IPT and GA-SQP algorithms as shown in Fig. 6 for th fin Problm. Th normal curv fitting tchniqu is usd to find how much th normal distribution accuratly fits th AEs of our proposd rsults of algorithms in ach cas of solvrs as shown in Fig. 6. It also displays th 95% confidnc intrvals (dottd curvs) for th fittd normal distribution. Fig. 6 shows that GA-IPT for m=30 is th bst among all algorithms and data is bst fittd with th Normal distribution. Furthr, Fig. 4 shows that GA-IPT for m=30 is bst among all algorithms rsults and also justifis th normal distribution of data. Ths confidnc lvls indicat that th prformanc of all mthods basd on th fittd normal distribution and GA-IPT showd high accuracy than th othr proposd mthods. It can b asily obsrvd from ths figurs that, th rsult obtaind by tchniqu GA-IPT in Fin is bttr than rsults obtaind by othrs algorithms. It is obsrvd that for m=30, our tchniqus show approximatly bttr rsults from rportd rsults and could potntially minimiz th rrors. 8

9 AE Conclusion In this sction, w xhibit our findings basd on numrical tchniqus carrid out in th prvious sction. A nw artificial intllignc approach is formulatd for solving an initial valu problm for rctangular fin using fd-forward Artificial Nural Ntworks, Activ st tchniqu, Intrior point Algorithm, Pattrn Sarch, Gntic Algorithm and hybridization PS-IPT. GA-AST, GA- IPT, GA-SQP. A complt statistical analysis is shown in Fig. for validation of proposd tchniqu. Comparison of th rsults is carrid out with a rfrnc solution calculatd through RK-4 Mthod, on th basis of which w hav concludd that our prsntd mthod has bttr ability of optimality than othrs numrical tchniqus as shown in Fig. 3 and Fig. 4. Th 3-dimnsional rprsntation of wights for two solvrs, AST and GA, is prsntd in Fig. 5. Th accuracy, convrgnc of th dsignd unsuprvisd ANN modl for hat transfr tmpratur distribution fin systm can b intnsifid by th us of rgnrat comptncy of intllignt tchniqu basd on Pattrn sarch, Gntic algorithm and its intgration with fficint local sarch tchniqu. Additionally, production of th strngth of optimization of volutionary computing through PS hybrid with local sarch mthodologis can b a good altrnativ for th improvd prformanc of th fin systm in trm of prformanc, dpndability, intllctual challngs and constancy. In th futur somon liks to us altrnat activation functions lik tangnt sigmoid, radial basis, Wavlts hat, Mxican hat tc., for daling with th hat transfr fin problm with bttr undrstanding. Fig 3: Shows th comparison btwn th rfrnc and th rportd rsult by diffrnt algorithms Comparison of AE.00E-09.00E-07.00E-05.00E-03.00E GA-30 AE AE (GA-AST) AE (GA-IPT) AE (GA-SQP) AE GA-45 AE GA-60 PS-IPT-30 Fig 4: Comparison of AE of diffrnt algorithms with diffrnt wights 9

10 Fig. 5: 3-Dimnsional rprsntation of ANN modl of wights straind with GA and AST Fig. 6: Shows th MAE, MSE and RMSE takn by 00 indpndnt runs. Rfrncs: [] Long, C. and. Sayma, N., Hat Transfr, Vntus Publishing, (p. 56), 009. [] Gurrum, S. P., t al., Thrmal issus in nxt-gnration intgratd circuits." IEEE Transactions on dvic and matrials rliability 4.4 (004), pp [3] Rmsburg R. Advancd thrmal dsign of lctronic quipmnt, Springr Scinc & Businss Mdia, 0. [4] McGln R. J., t al., Thrmal managmnt tchniqus for high powr lctronic dvics, Applid Thrmal Enginring, 4(8), (004): [5] Kraus, A. D., t al., Extndd surfac hat transfr. John Wily & Sons, 00. [6] Chang, M. H., A dcomposition solution for fins with tmpratur dpndnt surfac hat flux, Intrnational journal of hat and mass transfr, 48(9), (005), pp , 0

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